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Let-x-y-z-be-pairwise-distinct-real-numbers-if-x-y-7-z-x-y-24-y-z-7-x-y-z-24-z-x-7-y-z-x-24-Find-x-2-y-2-




Question Number 128204 by I want to learn more last updated on 05/Jan/21
Let   x,  y,  z   be pairwise distinct real numbers, if    (x  +  y  −  7)[z(x  +  y)  +  24]   =  (y  +  z  −  7)[x(y  +  z)  +  24]   =   (z  +  x  −  7)[y(z  +  x)  +  24]  Find     x^2   +  y^2   +  z^2
$$\mathrm{Let}\:\:\:\mathrm{x},\:\:\mathrm{y},\:\:\mathrm{z}\:\:\:\mathrm{be}\:\mathrm{pairwise}\:\mathrm{distinct}\:\mathrm{real}\:\mathrm{numbers},\:\mathrm{if} \\ $$$$\:\:\left(\mathrm{x}\:\:+\:\:\mathrm{y}\:\:−\:\:\mathrm{7}\right)\left[\mathrm{z}\left(\mathrm{x}\:\:+\:\:\mathrm{y}\right)\:\:+\:\:\mathrm{24}\right]\:\:\:=\:\:\left(\mathrm{y}\:\:+\:\:\mathrm{z}\:\:−\:\:\mathrm{7}\right)\left[\mathrm{x}\left(\mathrm{y}\:\:+\:\:\mathrm{z}\right)\:\:+\:\:\mathrm{24}\right]\:\:\:=\:\:\:\left(\mathrm{z}\:\:+\:\:\mathrm{x}\:\:−\:\:\mathrm{7}\right)\left[\mathrm{y}\left(\mathrm{z}\:\:+\:\:\mathrm{x}\right)\:\:+\:\:\mathrm{24}\right] \\ $$$$\mathrm{Find}\:\:\:\:\:\mathrm{x}^{\mathrm{2}} \:\:+\:\:\mathrm{y}^{\mathrm{2}} \:\:+\:\:\mathrm{z}^{\mathrm{2}} \\ $$

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